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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Representing measures and topological type of finite bordered Riemann surfaces


Author: David Nash
Journal: Trans. Amer. Math. Soc. 192 (1974), 129-138
MSC: Primary 30A48; Secondary 30A98
DOI: https://doi.org/10.1090/S0002-9947-1974-0385087-6
MathSciNet review: 0385087
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Abstract: A finite bordered Riemann surface $ \mathcal{R}$ with s boundary components and interior genus g has first Betti number $ r = 2g + s - 1$. Let a be any interior point of $ \mathcal{R}$ and $ {e_a}$ denote evaluation at a on the usual hypo-Dirichlet algebra associated with $ \mathcal{R}$. We establish some connections between the topological and, more strongly, the conformal type of $ \mathcal{R}$ and the geometry of $ {\mathfrak{M}_a}$ the set of representing measures for $ {e_a}$. For example, we show that if $ {\mathfrak{M}_a}$ has an isolated extreme point, then $ \mathcal{R}$ must be a planar surface. Several questions posed by Sarason are answered through exhausting the possibilities for the case $ r = 2$.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0385087-6
Keywords: Finite bordered Riemann surface, representing measure, real annihilating measure, hypo-Dirichlet algebra, convex body, extreme point, Betti number, interior genus, conformai equivalence
Article copyright: © Copyright 1974 American Mathematical Society